Exceptional points and pseudo-Hermiticity in real potential scattering

Abstract

We employ a recently-developed transfer-matrix formulation of scattering theory in two dimensions to study a class of scattering setups modeled by real potentials. The transfer matrix for these potentials is related to the time-evolution operator for an associated pseudo-Hermitian Hamiltonian operator $\hat{\mathbf{H}}$ which develops an exceptional point for a discrete set of incident wavenumbers. We use the spectral properties of this operator to determine the transfer matrix of these potentials and solve their scattering problem. We apply our general results to explore the scattering of waves by a waveguide of finite length in two dimensions, where the source of the incident wave and the detectors measuring the scattered wave are positioned at spatial infinities while the interior of the waveguide, which is filled with an inactive material, forms a finite rectangular region of the space. The study of this model allows us to elucidate the physical meaning and implications of the presence of the real and complex eigenvalues of $\hat{\mathbf{H}}$ and its exceptional points. Our results reveal the relevance of the concepts of pseudo-Hermitian operator and exceptional point in the standard quantum mechanics of closed systems where the potentials are required to be real.

26 pages, 3 figures

References in corpus (7)

• Physical realization of $\mathcal{P}\mathcal{T}$-symmetric potential scattering in a planar slab waveguide
• A non-Hermitian $PT-$symmetric Bose-Hubbard model: eigenvalue rings from unfolding higher-order exceptional points
• Exceptional Points in a Microwave Billiard with Time-Reversal Invariance Violation
• Geometric Phase for Non-Hermitian Hamiltonians and Its Holonomy Interpretation
• Exceptional Points in the Flatland: A Non-Hermitian Line-Wave Scenario
• Fundamental transfer matrix and dynamical formulation of stationary scattering in two and three dimensions
• PT symmetry breaking and exceptional points for a class of inhomogeneous complex potentials

Cited by in corpus (3)

• Non-standard quantum algebras and infinite-dimensional PT-symmetric systems
• Propagating-wave approximation in two-dimensional potential scattering
• Reciprocity Theorem and Fundamental Transfer Matrix